The grandparents, unfortunately, have to bear that burden & will probably have to become the baby's legal guardians & raise the child themselves.
If she abandons the child, yes.
no, the parents of the child have more rights to the child than the grandparents.
I assume that you mean "from the biologicalfather". The father have every right to have you see his child when the child stays with him. The mother can do nothing about that unless she can prove you are unfit or dangerous to the child and then she will need a court order to keep you away.
It depends on who has legal custody.
With her child, no.
If the grandparents are legal guardians and the child is living with them, the both of you as the biological parents have to pay child support to the grandparents.
Yes, if the case is reported by a social worker or if its brought to court by a close relative
She lived with her mother and (occasionally her father was there) in her grandparents house.
She can terminate her parental rights, not yours.
If the grandparents do not have legal permanent custody and the mother has proof she is the child's legal custodian, even if she has been absent in the child's life--yes, they can. If you believe the mother is unfit to care for the child or you have had physical custody for a long period of time with little or no contact from the mother, you can file an emergency appeal for custody with the courts. A temporary custody paper will not keep the child with you if she ever signed one as she can revoke it at anytime.
It's not for the mother to decide. He has to petition for visitation in court. She can not go against a court order.
With the facts given...the mother is the one that would qualify under the tests, especially the support and residence ones. Which without, the grandparents don't qualify to claim the child in any case.